I think you mean, in the first paragraph “the link of each ideal vertex is a torus” and in the second paragraph “the link of an ideal vertex is instead a surface of genus two”.

Such triangulations are sometimes called “pseudo-triangulations”, and the spaces they give “pseudo-manifolds”.

I think you mean, in the first paragraph “the link of each ideal vertex is a torus” and in the second paragraph “the link of an ideal vertex is instead a surface of genus two”.

Such triangulations are sometimes called “pseudo-triangulations”, and the spaces they give “pseudo-manifolds”. For a recent discussion see the paper Traversing three-manifold triangulations and spines by Rubinstein, Segerman, and Tillmann.